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My course notes list two reasons why cross-validation has a pessimistic bias. The first one is that the accuracy is measured for models that are trained on less data, which I understand. However, the second reason I don't understand. Supposedly, when we do cross validation and divide our data D into training sets D_i and test sets T_i, then the D_i and T_i are not independent (and even complementary) given D.

However, I don't see why this is different from the situation where we use a fixed testset: if we have a training set D and test set T, than T and D are also not independent given the union of D and T. In this case there is no bias, so I would expect there to be no bias for cross-validation either (apart from the fact that the model is trained on less data). Of course, since the different models that we train for cross validation use overlapping data, I would expect their accuracy to be correlated, which could lead to a higher variance, but I don't see how this could give a bias.

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  • $\begingroup$ I’ll just add that I agree with what you are saying. I don’t believe correlated training sets have anything to do with increased bias. In fact the common argument for the bias variance trade off in machine learning is that LOOCV has low bias but high variance due to the highly correlated training sets (LOOCV doesn’t actually have high variance necessarily but that’s another story) $\endgroup$
    – astel
    Nov 1 '20 at 3:30

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